Definitions

The volume flux through the curve between the points ${\displaystyle A}$ and ${\displaystyle P.}$

Lamb and Batchelor define the stream function ${\displaystyle \psi (x,y,t)}$ for an incompressible flow velocity field ${\displaystyle (u(t),v(t))}$ as follows.^[3] Given a point ${\displaystyle P}$ and a point ${\displaystyle A}$,

${\displaystyle \psi =\int _{A}^{P}\left(u\,{\text{d}}y-v\,{\text{d}}x\right)}$

is the integral of the dot product of the flow velocity vector ${\displaystyle (u,v)}$ and the normal ${\displaystyle (+{\text{d}}y,-{\text{d}}x)}$ to the curve element ${\displaystyle ({\text{d}}x,{\text{d}}y).}$ In other words, the stream function ${\displaystyle \psi }$ is the volume flux through the curve ${\displaystyle AP}$. The point ${\displaystyle A}$ is simply a reference point that defines where the stream function is identically zero. A shift in ${\displaystyle A}$ results in adding a constant to the stream function ${\displaystyle \psi }$ at ${\displaystyle P}$.

An infinitesimal shift ${\displaystyle \delta P=(\delta x,\delta y)}$ of the position ${\displaystyle P}$ results in a change of the stream function:

${\displaystyle \delta \psi =u\,\delta y-v\,\delta x}$.

From the exact differential

${\displaystyle \delta \psi ={\frac {\partial \psi }{\partial x}}\,\delta x+{\frac {\partial \psi }{\partial y}}\,\delta y,}$

the flow velocity components in relation to the stream function ${\displaystyle \psi }$ have to be

${\displaystyle u={\frac {\partial \psi }{\partial y}},\qquad v=-{\frac {\partial \psi }{\partial x}},}$

in which case they indeed satisfy the condition of zero divergence resulting from flow incompressibility, i.e.

${\displaystyle {\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}=0.}$

Derivation of the two-dimensional stream function

Consider two points A and B in two-dimensional plane flow. If the distance between these two points is very small: δn, and a stream of flow passes between these points with an average velocity, q perpendicular to the line AB, the volume flow rate per unit thickness, δΨ is given by:

${\displaystyle \delta \psi =q\delta n\,}$

As δn → 0, rearranging this expression, we get:

${\displaystyle q={\frac {\partial \psi }{\partial n}}\,}$

Now consider two-dimensional plane flow with reference to a coordinate system. Suppose an observer looks along an arbitrary axis in the direction of increase and sees flow crossing the axis from left to right. A sign convention is adopted such that the flow velocity is positive.

Continuity: the derivation

Consider two-dimensional plane flow within a Cartesian coordinate system. Continuity states that if we consider incompressible flow into an elemental square, the flow into that small element must equal the flow out of that element.

The total flow into the element is given by:

${\displaystyle \delta \psi _{\text{in}}=u\,\delta y+v\,\delta x.\,}$

The total flow out of the element is given by:

${\displaystyle \delta \psi _{\text{out}}=\left(u+{\frac {\partial u}{\partial x}}\delta x\right)\delta y+\left(v+{\frac {\partial v}{\partial y}}\delta y\right)\delta x.\,}$

Thus we have:

${\displaystyle {\begin{aligned}\delta \psi _{\text{in}}&=\delta \psi _{\text{out}}\,\\u\,\delta y+v\,\delta x&=\left(u+{\frac {\partial u}{\partial x}}\delta x\right)\delta y+\left(v+{\frac {\partial v}{\partial y}}\delta y\right)\delta x\,\end{aligned}}}$

and simplifying to:

${\displaystyle {\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}=0.}$

Substituting the expressions of the stream function into this equation, we have:

${\displaystyle {\frac {\partial ^{2}\psi }{\partial x\partial y}}-{\frac {\partial ^{2}\psi }{\partial y\partial x}}=0.}$

Vorticity

The stream function can be found from vorticity using the following Poisson's equation:

${\displaystyle \nabla ^{2}\psi =-\omega }$

or

${\displaystyle \nabla ^{2}\psi '=+\omega }$

where the vorticity vector ${\displaystyle {\boldsymbol {\omega }}=\nabla \times \mathbf {u} }$ – defined as the curl of the flow velocity vector ${\displaystyle \mathbf {u} }$ – for this two-dimensional flow has ${\displaystyle {\boldsymbol {\omega }}=(0,0,\omega ),}$ i.e. only the ${\displaystyle z}$-component ${\displaystyle \omega }$ can be non-zero.

Proof that a constant value for the stream function corresponds to a streamline

Consider two-dimensional plane flow within a Cartesian coordinate system. Consider two infinitesimally close points ${\displaystyle P=(x,y)}$ and ${\displaystyle Q=(x+dx,y+dy)}$. From calculus we have that

${\displaystyle {\begin{aligned}\psi (x+dx,y+dy)-\psi (x,y)&={\partial \psi \over \partial x}dx+{\partial \psi \over \partial y}dy\\&=\nabla \psi \cdot d{\boldsymbol {r}}\end{aligned}}}$

Say ${\displaystyle \psi }$ takes the same value, say ${\displaystyle C}$, at the two points ${\displaystyle P}$ and ${\displaystyle Q}$, then ${\displaystyle d{\boldsymbol {r}}}$ is tangent to the curve ${\displaystyle \psi =C}$ at ${\displaystyle P}$ and

${\displaystyle 0=\psi (x+dx,y+dy)-\psi (x,y)=\nabla \psi \cdot d{\boldsymbol {r}}}$

implying that the vector ${\displaystyle \nabla \psi }$ is normal to the curve ${\displaystyle \psi =C}$. If we can show that everywhere ${\displaystyle {\boldsymbol {u}}\cdot \nabla \psi =0}$, using the formula for ${\displaystyle {\boldsymbol {u}}}$ in terms of ${\displaystyle \psi }$, then we will have proved the result. This easily follows,

${\displaystyle {\boldsymbol {u}}\cdot \nabla \psi ={\partial \psi \over \partial y}{\partial \psi \over \partial x}+\left(-{\partial \psi \over \partial x}\right){\partial \psi \over \partial y}=0.}$

Existence of the stream function

Consider two-dimensional plane flow with velocity vector ${\displaystyle \mathbf {u} =(u,v,0)}$. Then ${\displaystyle \mathbf {u} }$ satisfies the curl-divergence equation:

${\displaystyle (\nabla \cdot \mathbf {u} )\,\mathbf {z} =-\nabla \times (R\,\mathbf {u} )}$

where ${\displaystyle \mathbf {z} =(0,0,1)}$ is a unit vector pointing in the positive ${\displaystyle z}$ direction, and ${\displaystyle R}$ is the ${\displaystyle 3\times 3}$ rotation matrix corresponding to a ${\displaystyle 90^{\circ }}$ anticlockwise rotation about the positive ${\displaystyle z}$ axis:

${\displaystyle R=R_{z}(90^{\circ })={\begin{bmatrix}0&-1&0\\1&0&0\\0&0&1\end{bmatrix}}.}$

If the flow is incompressible (i.e., ${\displaystyle \nabla \cdot \mathbf {u} =0}$) this gives

${\displaystyle \mathbf {0} =\nabla \times (R\,\mathbf {u} )}$ .

Then by Stokes' Theorem the line integral of ${\displaystyle R\,\mathbf {u} }$ over any closed loop vanishes (i.e., the line integral is path-independent). Finally, by the converse of the gradient theorem, a scalar function ${\displaystyle \psi }$ exists such that

${\displaystyle R\,\mathbf {u} =-\nabla \psi }$.

Here ${\displaystyle \psi }$ represents the stream function.

Conversely, if the stream function exists, then ${\displaystyle R\,\mathbf {u} =-\nabla \psi }$. Substituting this result into the curl-divergence equation yields ${\displaystyle \nabla \cdot \mathbf {u} =0}$.

In summary, the stream function for two-dimensional plane flow exists if and only if the flow is incompressible.

Properties of the stream function

1. The stream function ${\displaystyle \psi }$ is constant along any streamline.
2. The rate of change of stream function with distance is equal in absolute value to the velocity component perpendicular to the direction of change.
3. If two incompressible flow fields are superimposed, then the stream function of the resulting flow field is the algebraic sum of the two original stream functions.

Citations

Sources